AbstractWe give an example of a Banach spaceXsuch thatK(X, X) is not an ideal inK(X, X**). We prove that ifz* is a weak* denting point in the unit ball of Z* and ifXis a closed subspace of a Banach spaceY, then the set of norm-preserving extensionsH B(x* ⊗z*) ⊆(Z*,Y)* of a functionalx* ⊗Z* ∈ (Z⊗X)* is equal to the setH B(x*) ⊗ {z*}. Using this result, we show that ifXis anM-ideal inYandZis a reflexive Banach space, thenK(Z, X) is anM-ideal inK(Z, Y) wheneverK(Z, X) is an ideal inK(Z, Y). We also show thatK(Z, X) is an ideal (respectively, anM-ideal) inK(Z, Y) for all Banach spaces Z whenever X is an ideal (respectively, anM-ideal) inYandX* has the compact approximation property with conjugate operators.